Ok so basically all the inequality is working out "From which point do the terms i add on become less significant than 0.001" which also answers the original question. Then you solved that inequality to see that they become less significant that 0.001 when n> 9. Thats all it means.

Since this is an alternating series, each partial sum is BETWEEN the two previous sums. Yes, If you find a value of n such that the difference between two consecutive sums (which is just the value of the n th term) is less than 0.001, you know the error will be less than that.